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For distributed evolutionary dynamical systems of the “reaction-diffusion” and “reaction-diffusion-advection” types, we analyze the behavior of invariant numerical characteristics of the attractor as the diffusion coefficients decrease. We consider the phenomenon of multimode diffusion chaos, one of whose signatures is an increase in the Lyapunov dimensions of the attractor. For several examples, we perform broad numerical experiments illustrating this phenomenon.

An arbitrary diffeomorphism of an annular set of the form is considered, where is a ball in a Banach space and is a (finite- or infinite-dimensional) torus. A system of effective sufficient conditions is proposed which ensure that has a global attractor that can be represented as a generalized solenoid, that is, the inverse limit , where is an expanding linear endomorphism of the torus . Furthermore, the restriction is topologically conjugate to a shift map of the solenoid. Bibliography: 25 titles.

An arbitrary diffeomorphism f of class C1 acting from an open set $$\mathcal{U}\subset \mathbb{R}^{m}$$ , m ≥ 2, into $$f(\mathcal{U})\subset \mathbb{R}^{m}$$ is considered. Sufficient conditions for such a diffeomorphism to admit a hyperbolic mixing attractor are obtained.

We consider a system of two relay type differential-difference equations, with delay in the coupling between oscillators. The equations are coupled in a special way The system models an association of coupled neural oscillators. An important feature of the system is an additional delay in the connection chain. The delay allows obtaining new effects which is an essential complication of the system dynamics, and an appearance of coexisting special form attractors. We show that there coexist asymptotically orbitally stable solutions with summary 2n [n ϵ N) spikes on a period. Moreover, the first oscillator has m spikes, and the second one has 2n — m (m = 1, 2,..., 2n — 1) spikes on a period. We conclude that the additional delay leads to an accumulation of coexisting attractors with a given number of spikes on a period.

A model of neural association of three pulsed neurons with a delayed broadcast connection is considered. It is assumed that the parameters of the problem are chosen near the critical point of stability loss by the homogeneous equilibrium state of the system. Because of the broadcast connection the equation corresponding to one of the oscillators can be detached in the system. Two remaining pulsed neurons interact with each other and, in addition, there is a periodic external action, determined by the broadcast neuron. Under these conditions, the normal form of this system is constructed for the values of parameters close to the critical ones on a stable invariant integral manifold. This normal form is reduced to a four-dimensional system with two variables responsible for the oscillation amplitudes, and the other two are defined as the difference between the phase variables of these oscillators with the phase variable of the broadcast oscillator. The obtained normal form has an invariant manifold on which the amplitude and phase variables of the oscillators coincide. Dynamics of the problem is described on this manifold. An important result was obtained on the basis of numerical analysis of the normal form. It turned out that periodic and chaotic oscillatory solutions can occur when the coupling link between the oscillators is weakened. Moreover, a cascade of bifurcations associated with the same type of phase transformations was discovered, where a self-symmetric stable cycle alternately loses symmetry with the appearance of two symmetrical to each other cycles. A cascade of bifurcations of period doubling occurs with each of these cycles with the appearance of symmetric chaotic regimes. With further decreasing of the coupling parameter, these symmetric chaotic regimes are combined into a self-symmetric one, which is then rebuilt into a self-symmetric cycle of a more complex form compared to the cycle obtained at the previous step. Then the whole process is repeated. Lyapunov exponents were computed to study chaotic attractors of the system.

The models of association of two coupled neural oscillators with synaptic coupling are considered. An important feature of this association is an additional delay in the chain of connections. The system of relay differential-difference equations was chosen as the mathematical model of this association. The synaptic connection was modeled based on the modified idea of fast threshold modulation (FTM). The delay allows obtaining new effects which is an essential complication of the system dynamics, and an appearance of coexisting special form attractors. We show that there coexist asymptotically orbitally stable solutions with summary \(N\in {\mathbb {N}}\) spikes on a period. Moreover, the first oscillator has m spikes, and the second one has \(N-m\) (\(m=1,2,\ldots ,\,N-1\)) spikes on a period. We conclude that the additional delay leads to an accumulation of coexisting attractors with a given number of spikes on a period.

A certain special class of diffeomorphisms of an 'annulus' (equal to the Cartesian product of a ball in , , and a circle) is investigated. The so-called annulus principle is established, that is, a list of sufficient conditions ensuring that each diffeomorphism in this class has a strange hyperbolic attractor of Smale-Williams solenoid type is given.Bibliography: 20 titles.

In this paper, we consider so-called finite-dimensional flutter systems, i.e., systems of ordinary differential equations that come about, first, from the Galerkin approximation of certain boundary value problems of the aeroelasticity theory and, second, in a number of radiophysics applications. Small parametric oscillations of these equations in the case of 1: 3 resonance are studied. By combining analytical and numerical methods, it is found that this resonance can cause the hard excitation of oscillations. Namely, for flutter systems, there is shown the possibility of emerging, in parallel with the stable equilibrium zero state, both stable invariant tori of arbitrary finite dimension and chaotic attractors.

We introduce a mathematical model of a continual circular chain of unidirectionally coupled oscillators. It is a nonlinear hyperbolic boundary value problem obtained from a circular chain of unidirectionally coupled ordinary differential equations in the limit as the number of equations indefinitely increases. We study the attractors of this boundary value problem. Combining analytic and numerical methods, we establish that one of the following two alternatives takes place in this problem: either the buffer phenomenon (unbounded accumulation of stable periodic motions) or chaotic attractors of arbitrarily high Lyapunov dimensions.

Asymptotic and numerical analysis of relaxation self-oscillations in a three-dimensional system of Volterra ordinary differential equations that models the well-known Belousov reaction is carried out. A numerical study of the corresponding distributed model-the parabolic system obtained from the original system of ordinary differential equations with the diffusive terms taken into account subject to the zero Neumann boundary conditions at the endpoints of a finite interval is attempted. It is shown that, when the diffusion coefficients are proportionally decreased while the other parameters remain intact, the distributed model exhibits the diffusion chaos phenomenon; that is, chaotic attractors of arbitrarily high dimension emerge.

Some parabolic systems of the reaction-diffusion type exhibit the phenomenon of diffusion chaos. Specifically, when the diffusivities decrease proportionally, while the other parameters of a system remain fixed, the system exhibits a chaotic attractor whose dimension increases indefinitely. Various finite-dimensional models of diffusion chaos are considered that represent chains of coupled ordinary differential equations and similar chains of discrete mappings. A numerical analysis suggests that these chains with suitably chosen parameters exhibit chaotic attractors of arbitrarily high dimensions.

We suggest a phenomenological model for the development of turbulence in the form of the nonlinear Klein-Gordon equation perturbed by nonconservative disturbances. Combining analytic and numerical methods, we establish that the transition to turbulence in this equation can follow both the Landau and the Landau-Sell scenarios. As is known, the first scenario is related to a cascade of bifurcations of stable invariant tori of increasing dimensions. The other scenario is related to a chaotic attractor whose Lyapunov dimension increases indefinitely under variation of a certain control parameter.

We study the problem of attractors of the two-dimensional mapping $$(u,v) \to (v, - (1 - \mu )u - F(v)),F(v) = \left\{ {\begin{array}{*{20}c} {q_1 forv > 0,} \\ {0forv > 0,} \\ { - q_2 forv > 0,} \\ \end{array} } \right.$$ where 0 0. This mapping is the mathematical model of a self-excited oscillator with relay amplifier and a part of the long transmission line without distortions in the feedback circuit. We prove that, in the system under study, there coexist stable cycles with arbitrarily large periods as the parameter µ decreases properly. We also show that the total number of these cycles increases without bound as µ → 0, i.e., the buffer phenomenon is realized.

A method is proposed for designing chaotic oscillators. Mathematically, three so-called partial oscillators S j (j = 1, 2, 3) are chosen, each of which is modeled by a nonlinear system of ordinary differential equations with a single attractor—an equilibrium or a cycle (the case S 1 = S 2 = S 3 is not excluded). It is shown that, when unidirectionally connected in a circle of the form with suitably chosen parameters, these oscillators can exhibit a joint chaotic behavior.

The buffer phenomenon is established for some classical mechanics problems that are described by pendulum-type equations with time-periodic small additive terms. This phenomenon is as follows: the systems under consideration can have an arbitrary fixed number of stable periodic modes if the system parameters are properly chosen.